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I am interested in the correspondence of algebraic results about C(X) (the space of continuous functions $X\to {\mathbb C}$(complex numbers) or $X\to {\mathbb R}$(real numbers) and topological properties of X,for example results like this .(Does by the way someone know what's the "deep result by bkouche" mentioned in the paper?)

Can you by the way use this result to prove interesting theorems with this translation
(like: -a manifold(or even a CW-complex) is paracompact
-the theorem of Tietze, etc?)

There are many such correspondences which are obtained by using Gelfand-Naimark but I couldn't find literature where you can find full details with all needed definitions and proofs.(I couldn't even find a proof of the categorical Gelfand-Naimark theorem in the nonunital case,only some sketches.) Does such literature exist for a beginner in this topic? The book "Basic Noncommutative Geometry" written by Khalkhali is a good source but omits most details (see page 16 for a little list).

So I would be glad if you can recommend to me a good book/link, or write a nice result here if it's not too complicated.

Example: X is connected iff C(X) has no idempotents because direct sums of subalgebras correspond to disjoint union of closed subspaces,since C is an equivalence.

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I took the liberty of cleaning up the formatting in your question and correcting some language errors. Hope you don't mind. – Yemon Choi Jan 16 '11 at 20:24
Thank you very much! – trew Jan 16 '11 at 20:36
"$X$ connected <=> $C(X)$ has only trivial idempotents" is trivial from the sheaf property and does not require such a deep Theorem as Gelfand-Naimark. – Martin Brandenburg Jan 17 '11 at 8:48
I've written up the proof of Gelfand-Neimark in the non-unital case here – Martin Brandenburg Jan 17 '11 at 8:49
sorry,i forgot about your article ;) – trew Jan 17 '11 at 8:57

Dear trew, here is an application I like of Gel'fand-Naimark's representation theorem . I'm not sure it answers your question in a strict sense but I hope it is sufficiently close... .

Consider a completely regular topological space $X$ and its algebra of continuous and bounded functions $A= \mathcal C_b (X) $. By Gel'fand-Naimark's representation theorem, the algebra $A$ is isomorphic to to $ \mathcal C (\bar X)$ for a uniquely (up to homeomorphism ) defined topological space $\bar X$. Well, this $ \bar X$ is the Stone–Čech compactification $\bar X=\beta (X)$ of $X$. There are other definitions of that compactification but I find this one appealing to those who (like me) are more familiar with spectra of rings than with ultrafilters.

Technical note Completely regular (= $T \;\; 3\frac{1}{2}$) means Hausdorff and global continuous functions ( not necessarily bounded ! ) numerous enough to separate a closed set from an exterior point. This is needed in order that $X$ embed into $\bar X=\beta (X)$.

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Thank you!I know this from the book of pedersen.Its a very nice application and there is even a one-to-one corresponende of the compactifications of a locally compact space and essential extensions of $C_0 (X) $ described in the book of Khalkhali. – trew Jan 16 '11 at 21:54
Thank you for this nice piece of information, trew : you obviously know more than I do on the subject! But I'll leave my answer for the benefit of users who might not have seen this application before. – Georges Elencwajg Jan 16 '11 at 22:10

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